Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Entropy-dissipating semi-discrete Runge–Kutta schemes for nonlinear diffusion equations

Pages: 27 – 53

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n1.a2

Authors

Ansgar Jüngel (Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria)

Stefan Schuchnigg (Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria)

Abstract

Semi-discrete Runge–Kutta schemes for nonlinear diffusion equations of parabolic type are analyzed. Conditions are determined under which the schemes dissipate the discrete entropy locally. The dissipation property is a consequence of the concavity of the difference of the entropies at two consecutive time steps. The concavity property is shown to be related to the Bakry–Emery approach and the geodesic convexity of the entropy. The abstract conditions are verified for quasilinear parabolic equations (including the porous-medium equation), a linear diffusion system, and the fourth-order quantum diffusion equation. Numerical experiments for various Runge–Kutta finite-difference discretizations of the one-dimensional porous-medium equation show that the entropy-dissipation property is in fact global.

Keywords

entropy-dissipative numerical schemes, Runge–Kutta schemes, entropy method, geodesic convexity, porous-medium equation, Derrida–Lebowitz–Speer–Spohn equation

2010 Mathematics Subject Classification

65J08, 65L06, 65M12, 65M20

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Published 10 January 2017